Fischer inner product ($⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$)

The Fischer inner product on polynomials is $⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$. Multiplication by $x_i$ is adjoint to differentiation by $x_i$. Therefore multiplication by $|x{|}^2$ is adjoint to the ordinary Laplacian, yielding the harmonic decomposition of homogeneous polynomials by finite-dimensional orthogonality.